Are the fans on even when the ignition switch is at Off? If so, then this could be a relay with a bad internal contact that is remaining closed. (If both fan motors are running, the sub-fan relay remaining closed could cause this. If the sub-fan relay is faulty, and is pulled out, both fans should stop.)
If the fans are not on with the switch at Off, but come on as soon as the switch is at On, do they run even when the instrument panel temperature gauge is below the normal position (~9:00 o'clock)? If they do run when the gauge is below normal, then the cause is not likely to be the engine coolant temperture sensor. In this case, what engine and transmission does your 2005 have? There was a recall issued back in 2007 relating to a faulty fan relay that could damage the ECM, and cause the fans to remain on. (See attached.)
If the fans run continuously, but only when the engine is also running, then check the HVAC controls. If the controls are set to either of the two windshield defrost positions and the HVAC fan control is not at Off, the AC can be running (even though AC hasn't actually been turned on) and when it does, the two radiator fans will usually be running as well.
Relays can be checked for continuity through the energizing coil, and for continuity through the internal switching contacts when the relay is energized. There's three relays involved in the control of the two fan motors. According to the recall notice, one of the relays, main relay #2, was defective. On page 4 of the recall notice there's a photo of what the replacement relay looks like. Check to see if your relay has the same reference number and the gray dot.Hi, thanks for the detailed reply. It's exactly as in your paragraph 2 - both fans are always on as soon as the switch goes on, even when completely cold. A/C compressor off. However, it's not a turbo (could use it) and so doesn't seem to fit the parameters of the recall notice? Should I check the relays anyway? Is there a procedure for that that you know of? I have meters. Thanks.